How do I fit label precisely around bottle, specifically the circumference?
To unwrap a cylinder in correct proportion:


Cut a vertical seam where you want it
Set an orthogonal view (poles or equator.. here, equator)
In edit mode, U unwrap  with 'Cylinder Projection' and settings as shown..





Oddly, in the 'Radius' field, entering pi*radius


.. which  doesn't make sense to me .. but there it is...